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Dividend Yield and Cost of Carry in European Call Pricing

Article Quant Q&A · Author: quis est ille
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Summary

This note addresses the European call formula for a dividend-paying stock and whether an expected stock growth rate should replace dividend yield. It explains that the dividend-adjusted Black–Scholes formulation and a cost-of-carry formulation are equivalent when their inputs are related consistently: cost of carry equals the risk-free rate minus the dividend yield. Thus the carry term can also be viewed as negative expected growth under the corresponding no-arbitrage setup.

The answer resolves the apparent disagreement as a difference in parameterization rather than a failure of no-arbitrage pricing. A forecast growth rate chosen independently of the dividend yield and risk-free rate represents a different assumption and will not preserve that equivalence. The note offers no derivation, empirical test, or treatment of market frictions; its conclusion depends on consistent model inputs and the generalized Black–Scholes framework.

Key ideas

  • Dividend-adjusted Black–Scholes pricing uses the stock’s dividend yield in the carry adjustment.
  • A cost-of-carry formulation is equivalent when carry is defined consistently with the risk-free rate and dividend yield.
  • An independently forecast growth rate does not generally match the no-arbitrage dividend-adjusted formulation.
  • The equivalence is presented within a generalized Black–Scholes setup and does not analyze market frictions.

Tags

OptionsEquitiesDerivatives pricing
Full text
# European option on a dividend paying stock, limits to arbitrage?


# European option on a dividend paying stock, limits to arbitrage?












What is the price C of a European call option on a dividend paying stock? I believe it is:

C = U. N(d1) - exp(-rt).K.N(d2)

d1 = [ ln(U/K) + (r + v^2/2).t ]/[ v.sqrt(t) ]

d2 = d1 - v.sqrt(t)

U = S.exp(-qt)

Where S is the spot price of the stock, q the dividend yield, K the strike, r the risk free rate, t time to expiry, v implied vol, and N the cumulative normal distribution function. All yields/rates continuous basis.

However, I have been told that because there is no long-dated forward price for stocks, the no arbitrage principle fails to apply, and so in this case the correct formula for U is

U = S.exp( (g – r)t )

where g is the forecast growth rate of the stock. Since q is typically 2-3%, and since g is assumed to be 5%, this leads to a considerable difference in the two methods. Which is correct?

[Edit] In the interests of clarity (see the first answer below, which claims the formulas are really the same) note the inconsistent assumptions about growth in the original question. If q = 3%, and r = 2.5%, this would imply a negative growth rate g of -0.5% in a no arbitrage world. However, it has been argued that where no arbitrage does not apply, we can use a forecasting model that predicts 5% growth in equities in the long term, and so g-r <> -q.

## Answer by SmallChess (score -1)

https://quant.stackexchange.com/a/21642

Both formulas (generalised Black Scholes) are correct. In the first one, you have the BS model with dividend. In the second, you have a non-zero cost of carry. It's really the same formula. Note that cost of carry = risk free rate - dividend. You can think cost of carry like the negative growth rate.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.